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anyanavicka [17]
3 years ago
12

Solve for x. Determinants

Mathematics
1 answer:
joja [24]3 years ago
6 0

Remember that row operations don't affect the determinant:

\begin{vmatrix}x+a&x&x\\x&x+a&x\\x&x&x+a\end{vmatrix}=\begin{vmatrix}a&a&0\\0&a&a\\-a&0&a\end{vmatrix}

(that is, subtract row 2 from row 1; subtract row 3 from row 2; and subtract row 1 from row 3. This is not the only way to do it, of course.)

Factoring out a gives

a^3\begin{vmatrix}1&1&0\\0&1&1\\-1&0&1\end{vmatrix}

Notice that adding rows 1 and 3 gives the same numbers in row 2. In other words, the rows are not linearly independent, which means the matrix is singular, and this is so for any value of x.

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